[/quote]
,
and 
Prove that
is given. A cube with side length 
is made up of 
unit cubes, each of which is either black or white. It turns out that among any 
cubes that share a common vertex and form a 
cube, there are at most 
black cubes. What is the maximum number of black cubes that could have been used?
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
Prove that
-
,
,
,
according to the following rule: If 
,
,
if 
(taking 
)
is the symbol other than 
if 
.
for which there exists some integer 
such that 
.
,
,
?
the sum of all positive divisors of 
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
,
,
is the point on 
such that 
.
be the circle through 
tangent to 
at 
the circle through 
at 
.
be the second intersection of 
lies on 
.
Prove that
Y by Yrock
Summary of the problem: When using exit() (or quit()) in the Python windows on AoPS the "Description" and "To fix" options show up as "undefined"
Page URL: N/A
Steps to reproduce:
1. Use the AoPS Python module to execute the exit() or quit() functions in a program. (try running the sample program)
Expected behavior: The "Description" and "To fix" sections give a description of SystemExit
Frequency: Always
Operating system(s): Windows 11 Home
Browser(s), including version: Microsoft Edge 130.0.2849.46
Additional information: N/A
Page URL: N/A
Steps to reproduce:
1. Use the AoPS Python module to execute the exit() or quit() functions in a program. (try running the sample program)
Expected behavior: The "Description" and "To fix" sections give a description of SystemExit
Frequency: Always
Operating system(s): Windows 11 Home
Browser(s), including version: Microsoft Edge 130.0.2849.46
Additional information: N/A
Attachments:
This post has been edited 2 times. Last edited by SoaringHigh, Oct 22, 2024, 2:11 PM
